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Abstract. -Continuous time random walk (CTRW) is studied with a new dynamic equation based on the age-structure of states. For a CTRW in a closed molecular system, two necessary conditions for microscopic reversibility are introduced: 1) independence of transition direction and waiting time for every state and 2) detailed balance among the transition probabilities. Together they are also sufficient condition. For a CTRW in an open system with explicit chemical energy input 1) still holds while 2) breaks down. Hence, CTRW models not satisfying 1) are either inconsistent with thermodynamics or cannot attain equilibrium due to hidden dissipation in non-Markovian states. Each CTRW defines a unique corresponding Markov process (cMP). The steady-state distribution of a CTRW equals that of the corresponding Markov process, and the two systems have the same steady-state flux, the same exit probabilities and the same mean trapping times. Mechanicity is discussed; a paradox observed by Kolomeisky and Fisher (J. Chem. Phys., 113 (2000) 10867) is resolved.Stochastic models are the theoretical basis for describing dynamics of biological, chemical and physical processes at the single-molecule level [1,2]. Recent studies in single-molecule spectroscopy and enzymology have invigorated a classic subject, the continuous time random walks (CTRW), which was developed and extensively studied many years ago by Montroll and coworkers [3,4]. See [5][6][7] for the recent work that motivated CTRW in connection to single motor proteins and single-enzyme kinetics.Applying stochastic models to molecular procsses requires serious considerations of the microscopic reversibility. When a closed molecular system with fluctuations reaches its stationary state, it is necessarily a chemical equilibrium with zero flux in any part of the system. This realization led to the introduction of detailed balance (DB) as a necessary condition for Markov kinetic models of closed system [8]. The celebrated fluctuation-dissipation relation [9] is in fact an explicit expression of detailed balance in the Langevin dynamics [10]. When a molecular system is in an open environment with sustained chemical input and dissipation, detailed balance breaks down [2, 11], which gives rise to free energy transduction, such as in c EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx

Molecular motors operate in an environment dominated by thermal fluctuations. A molecular motor may produce an active force at the reaction site to directly move the motor forward. Alternatively a molecular motor may generate a unidirectional motion by rectifying thermal fluctuations. In this case, the chemical reaction establishes free energy barriers to block the backward fluctuations. The effect of the chemical reaction on the motor motion can be represented by the motor potential profile (rectifying barrier andor active driving force). Different motor mechanisms are characterised by different motor potential profiles. The mathematical theory and properties of molecular motors are discussed and a mathematical framework is developed for extracting the motor potential profile from measured time series of motor position. As an example, we discuss the binding zipper model for the F(1) ATPase, which was motivated mainly by the fact that the motor potential profile of the F(1) ATPase is nearly a constant slope.